Pricing electricity forward contracts under observable information … · Japan Electric Power Exchange: JEPX ¾From short term (half hour) to long term (1 week—1 year) 3 ¾Match

AJRC and RIETI workshop on economic and financial analysis of commodity markets

September 14—15, 2017

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Pricing electricity forward contracts under observable information in JEPX

(Japan Electric Power Exchange)

Faculty of Business SciencesUniversity of Tsukuba, Tokyo, Japan

E-mail: [email protected]://www2.gssm.otsuka.tsukuba.ac.jp/staff/yuji/

Y. Yamada

Acknowledgement

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Supported by Grant-in-Aid for Scientific Research (A) 16H01833 (PI: Yuji Yamada) from Japan Society for the Promotion of Science (JSPS).

(a)Faculty of Business Sciences, University of Tsukuba(b)Faculty of Science and Technology, Tokyo University of Science(c)Faculty of Engineering Division I, Tokyo University of Science(d)Faculty of Science and Engineering, Chuo University

Naoki Makimoto(a), Setsuya Kurahashi(a)

Ryuta Takashima(b), Nobuyuki Yamaguchi(c)

Junya Goto(d)

Collaborators

Japan Electric Power Exchange: JEPX

From short term (half hour) to long term (1 week—1 year)

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Match sell and buy orders for delivering a fixed amount of electricity (kWh) in nine areas, Hokkaido, Tohoku, Tokyo, Chubu, Hokuriku, Kansai, Chugoku, Shikoku, and Kyushu

(http://www.jepx.org/)

Forward contractSpot contract

Japan Electric Power Exchange: JEPX

Spot electricity in JEPX: Every half hour, 48 products a day.Orders are closed at 9am 1 day before the delivery.

(http://www.jepx.org/)

Forward contracts:Unique spot price in the entire 9 areas of Japan

Written on the system price

Illiquid, only traded several times a year!

System price

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Outline

Forward prices under observable information

1. Forward contracts in JEPX, its expression using daily forwardprices, and an application of Esscher transform

2. Derivation of daily forward price via Esscher transformation

Comparison with Girsanov transformation

Modeling of underlying spot price using state space equations

3. Empirical simulation

4. Concluding remarks

(JEPX spot price)

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Financial settlement between forward and system prices for the delivery period from 1 week to 1 year in the future.

Forward contracts in JEPX

: Average system price at day (spot price) , : Forward price with days delivery starting at day (> )

, , ,Float CF

Issue date

Fixed CF

Total payoff: , = × ,

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Expression using daily forward prices, : = , : Daily forward price of delivering at day only

Total payoff: , = ,

, = 1 ,

= × ,No arbitrage!

, , ,Float CF

(Stochastic)

Float CF(Deterministic)

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Spot and daily forward prices, : = , : Daily forward price of delivering at day only

Payoff:

Connection between spot and forward prices

( ): Deterministic function of

- Seasonal and long term trends + Day-of-the-week effects

ln = + = exp +: Spot price process on , , ,

( ): -adopted stochastic process (observed at time )

, Function of

Forward price via Esscher transform (Kijima and Tanaka‘07)

: = ,Change of measure:

Forward price:

,, =

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Esscher transform: = exp + Uncertain random variable

: Esscher paramter

= 1:=

= 1

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Forward price and risk premium

, = 1 ,Forward price with days delivery:

Risk premium Seller > Buyer Seller < Buyer Seller = Buyer

Price

may be implied by matching with the realized forward price.

= 0 (i. e. , = 1) is risk neutral so that , = .

Positive (negative) reflects on the sellers’ (buyers’) risk premium.

, = = 1 = 1

= 0Risk neutral

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Outline








(JEPX spot price)

Modeling of underlying spot price

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( ): Seasonal and long term trends + Day-of-the-week effects

ln = + = exp += , = +

: state variable, : white noise, , × , × , : constant matrices

Observed equation: State equation:

( ): -adopted stochastic process (observed at time )

- , e.g., Durbin & Koopman ‘01

e.g., Gibson & Schwartz ‘90, Schwartz & Smith ‘00

- Multi-factor model

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Computation of forward price

, = = ==, Conditionally independent of

, = , ,,= exp + , + , + 1 ,

, : Cumulant generating function

- , : Predicted value of .

= 1

, : Prediction error.

= exp + == + = +

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- Girsanov:


= , : Market price of risk

under + under

- Esscher: = , : Esscher parameter

under + under

: -dimensional Brownian motion

Continuous time model: = + + , =

= ( )Equivalence condition:

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= + += = + ( ) + ( )

Girsanov transform

= = ( )( ) =

Esccher transform

Proof: = + + , =Comparison with Girsanov transformation

= ( )Equivalence condition:

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Outline








(JEPX spot price)

Step 2) Apply ( ) for { } and compute distribution of prediction errors , .

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Step 1) Estimate trend function and residual using underling spot price data , = 1,… , .

Empirical simulation

, = , ,,

- , = : Predicted value of . , : Prediction error.

Spot price (daily average of 48 system prices released every day)Data period: 2012/4/1 2016/12/31 ( = 1736)

Daily forward price (Esscher):

Step 3) Comparison with realized forward prices.

Step 1) Estimation of seasonal trend using GAM

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: Holiday dummy, : Day dummy (0 or 1)

: Seasonal trend dummy (e.g., 1,…,365 (366))

,0),...,365(364tSeasonal)366(365,...,1tSeasonal

).731(730,...),367(366tSeasonal

Choose spline function for = 1,… , 365 (366).

Generalized Additive Model (GAM; Hastie and Tibshirani‘90)

ln = + × + × + +

Yamada et al. ’14: Apply GAM with repetitive data with

Estimation of trend function by GAMs

ln = + × + × + +

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(Mon—Sat and Holiday) Estimated seasonal trend function

Data period: 2012/4/1 2016/12/31 ( = 1736)

Predicted values for { } using ( ) model with = :

: Current date, : Learning data period, : prediction horizon

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Learning data ( days) Predict

Compute prediction error

Repeat until day

Start ( )

Given learning data,

Step 2) Estimation of prediction errors

, = , × ,, , = , … ,Daily forward price using empirical distribution:

+ 1construct ( ). Predict by , =

and compute , for = ,… , .

Mean Absolute Error (left) and Standard Deviation (right)

Prediction errors of ( ) when = 90 days:

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= 1 + 1 , , = 1 , ,

Term structure of prediction errors

Empirical estimation of daily forward prices

MAE between the daily forward pricewith = 0 and spot price

= ,, : Risk premium rate

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, = , × ,, , = , … ,Estimated value of daily forward price:

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Step 3) Comparison with realized forward prices

, = 1 ,Estimated forward price:

Realized forward prices in JEPX (2012/4/1 2016/12/31):

Year of transaction 2012 2013 2014 2015 20161 week delivery 0 0 8 2 (1)1 month delivery 0 0 2 3 -

888 2 (111)))))))))))))))))))))))) 11 forward prices


Price= 0

Risk neutral

may be implied by matching with the realized forward price.

Risk premium of realized forward price?Estimated sample path of forward price with implied ?

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Step 3) Comparison with realized forward prices

Forward payoff:

= , = 1 1 ,,Payoff normalized by

number of days

Predicted vs. realized values of the underlying vs. forward payoffs?


Price= 0

Risk neutralPredicted

value of the underlying

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Predicted and realized values vs. forward payoff

“Realized Predicted of the underlying”& Normalized forward payoff Implied by AR and RW predictions

- Realized payoff is significantly negative.AR RW Payoff

0.427 0.498 0.918**

MAE 0.770 1.18 1.10 - Implied by AR prediction is always positive.

Case with maximum payoff size

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AR prediction RW prediction

- In the case of RW prediction, the forward price fluctuates with the daily spot price and is affected by “spikes.”

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Case with minimum payoff size

AR prediction RW prediction

- In the case of AR prediction, the forward price is smoothand seems to reflect on the long term average of spot prices.

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Concluding remarks

In the case of RW prediction, the forward price fluctuates with the daily spot price and is affected by “spikes.”

Generalized additive model (GAM)

In the case of AR prediction, the forward price is smoothand seems to reflect on the long term average of spot prices.

Forward prices under observable information (JEPX spot price)

Empirical simulation

Modeling of spot price dynamics and forward prices

- Trend functionState space equation formula- Stochastic partEsscher transformation- Forward price

Documents

Pricing electricity forward contracts under observable information … · Japan Electric Power Exchange: JEPX ¾From short term (half hour) to long term (1 week—1 year) 3 ¾Match